I'm wondering if there is any way to prove that of the numbers in the form 1.....1 (i.e. all digits are 1), 1 is the only cube number.
first we use the fact that .
proof: using fermat's li'l theorem so either or . the latter implies .
so if has to be a cube then the inly admissible values of would be the ones satisfying . there are only trials needed to know which are those. the possible values of are . that's it.
in my last post i missed the numbers of the type .